Temperature data option calculation

Author
Affiliation

Pietro Rota

Università cattolica del sacro cuore

Modified

May 17, 2025

1 Weather Derivatives Temperature Options

Full dependencies list, click to expand list 
[1] "R version 4.4.3 (2025-02-28 ucrt)"
[1] "x86_64-w64-mingw32/x64"

Functions loaded from main

Functions.1 Functions.2 Functions.3
check_acc desc_df find_outliers
normalize quickplot remove_outliers
RSS show_df smart_round

Packages required to run this file

Show code 
required_packages(file)
Required.Packages.1 Required.Packages.2 Required.Packages.3
caret changepoint dplyr
DT e1071 fBasics
forecast gganimate ggplot2
gt gtExtras leaflet
lubridate MASS minpack
nlme nlstools NMOF
PerformanceAnalytics plotly quantmod
reshape2 rugarch sde
splines stats4 tidyr
timeDate timeSeries TTR
xts zoo

Functions defined in this document

Required.Functions.1 Required.Functions.2 Required.Functions.3
apply_convolution sin_component create_spline_plot
create_poly_plot fourier_series create_fourier_plot
create_changepoint_plot model model_fit
T_model dT_model spline_fit
euler_step monte_carlo_temp model_formula
Show code 
cat("time of creation", "\n")
time of creation
Show code 
print(file.info(file)$ctime, "\n")
Warning in as.POSIXlt.POSIXct(x, tz): unknown timezone '
'
[1] "2025-05-06 15:32:44 GMT"
Show code 
cat("LAST MODIFICATION", "\n")
LAST MODIFICATION
Show code 
print(file.info(file)$mtime, "\n")
Warning in as.POSIXlt.POSIXct(x, tz): unknown timezone '
'
[1] "2025-05-16 22:38:07 GMT"
Show code 
cat("Last Access", "\n")
Last Access
Show code 
print(file.info(file)$mtime, "\n")
Warning in as.POSIXlt.POSIXct(x, tz): unknown timezone '
'
[1] "2025-05-16 22:38:07 GMT"

References:

  1. Weather Futures and Options (CME Group Inc, 2021) [LINK]

  2. Managing Climate Risk with CME Group Weather Futures and Options (Dominic Sutton-Vermeule, 20-Jan 2021) [LINK]

  3. MANAGING CLIMATE RISK IN THE U.S. FINANCIAL SYSTEM, ISBN: 978-0-578-74841-2 (2020) [LINK]

2 Climate data download

Much of the weather market is dominated by temperature derivatives, aimed to protect the holder from unexpected temperature prints. The underlying used to trade these contracts is either Heating Degree Days (HDD) or Cooling Degree Days (CDD). For a specific day n, these can be defined as:

The most popular temperature instruments traded are options, whose payoff function depends on a cumulative sum over a longer period, usually an entire season:

\(HDDn = \max(T_{ref} - T_n, 0)\)

\(CDDn = \max(T_n - T_{ref}, 0)\)

Calls protect you from extreme circumstances, meanwhile puts protect you from mild climates
Option Type Protection Against Exercise When Payout Example
HDD call Overly cold winters HDD > K \(\alpha \cdot\) (HDD - K) Farmers
HDD put Overly warm winters HDD < K \(\alpha \cdot\) (K - HDD) Ski resorts
CDD call Overly hot summers CDD > K \(\alpha \cdot\) (CDD - K) Utilities
CDD put Overly cold summers CDD < K \(\alpha \cdot\) (K - CDD) Beaches

NASA Prediction Of Worldwide Energy Resources (POWER) | Data Access Viewer (DAV) [LINK]

This dataset is from the Agroclimatology community, which is crucial for ensuring reproducibility in climate research.

This study utilizes temperature data obtained from the MERRA-2 (Modern-Era Retrospective analysis for Research and Applications, Version 2) dataset, which provides detailed meteorological measurements at a spatial resolution of 0.5 x 0.625 degrees latitude/longitude.

The specific region analyzed, with an average elevation of 288.19 meters, is located at a latitude of 45.5330, and longitude of 9.1911.

Show code 
library(leaflet, quietly = TRUE, warn.conflicts = FALSE)
map_data <- data.frame(
  name = "Location",
  lat = 28.3688,
  lon = -81.5614
)
# Create a leaflet map
leaflet(map_data) %>%
  addTiles() %>%  # Add default OpenStreetMap tiles
  addMarkers(~lon, ~lat, label = ~name, ) %>%
  addCircleMarkers(~lon, ~lat, radius = 10, color = "red", fillOpacity = 0.8) %>% 
  setView(lng = map_data$lon, lat = map_data$lat, zoom = 12)

It is important to note that the dataset uses a value of -999 to denote missing data, which I’ve removed and replaced with the mean of the overall dataset. Either when a parameter cannot be computed or falls outside the range of available source data

Parameter(s):

  • T2M_MAX = Temperature at 2 Meters Max for the day (C)

  • T2M_MIN = Temperature at 2 Meters Min for the day (C)

  • T2M_AVG = \(\frac {T_{MIN}+T_{MAX}}2\) Temperature at 2 Meters Average for the day (C)

Temperature at 2 meters refers to the air temperature measured 2 meters (approximately 6.5 feet) above the ground. This is the standard height for temperature measurements used in meteorology because it minimizes the effects of ground heating or cooling, providing a more accurate reflection of the ambient air temperature experienced by humans and relevant to various economic activities.

Why? Temperature at 2 meters is commonly used as the basis for weather derivatives. This is because:

  • Standardization: Temperature at 2 meters is a standardized and widely available data point, making it reliable for use in financial contracts.

  • Relevance: Many weather-related risks, such as heating degree days (HDD) and cooling degree days (CDD), are calculated based on temperatures at this height. These metrics are commonly used in weather derivatives to hedge against risks related to heating and cooling needs.

  • Accuracy: Because it represents the temperature most relevant to human activities and energy consumption, it is directly applicable for creating and settling weather derivatives that are based on temperature-related indices.

Show code 
ORIGINAL_DATASET <- read.csv("DISNEY DATA.csv", skip = 10)

DATASET <- ORIGINAL_DATASET %>% 
  mutate(T_MAX=remove_outliers(T2M_MAX, fill = "NA") %>% na.approx) %>% 
  mutate(T_MIN=remove_outliers(T2M_MIN, fill = "NA") %>% na.approx) %>% 
  mutate(DAY = as.Date(ORIGINAL_DATASET$DOY - 1, origin = paste0(ORIGINAL_DATASET$YEAR, "-01-01"))) %>%
  mutate(Month=month(DAY)) %>% 
  dplyr::select(DAY, YEAR, Month, DOY, T_MAX, T_MIN)
  

DATASET$T_AVG <- apply(DATASET[c("T_MAX", "T_MIN")], 1, mean)

ORIGINAL_DATASET[ifelse(find_outliers(ORIGINAL_DATASET$T2M_MAX)==0,FALSE,TRUE),] %>% 
  group_by(YEAR) %>% 
  summarise("Days" = length(DOY)) %>% 
  ggplot( aes(x = YEAR, y = Days))+
    geom_bar(stat = "identity") + 
    labs(x= NULL, y =NULL, title = "Days where the sensor malfunctioned", 
        subtitle = "Identified by remove outliers")

Show code 
datatable(smart_round(DATASET, digits = 3))
Warning: There was 1 warning in `mutate()`.
ℹ In argument: `across(where(is.numeric), round, digits = digits)`.
Caused by warning:
! The `...` argument of `across()` is deprecated as of dplyr 1.1.0.
Supply arguments directly to `.fns` through an anonymous function instead.

  # Previously
  across(a:b, mean, na.rm = TRUE)

  # Now
  across(a:b, \(x) mean(x, na.rm = TRUE))
Warning in instance$preRenderHook(instance): It seems your data is too big for
client-side DataTables. You may consider server-side processing:
https://rstudio.github.io/DT/server.html

2.1 Initial data visualization

Show code 
cleandataset <- DATASET %>% 
  select(T_MAX, T_MIN, T_AVG) %>% 
  xts(order.by = DATASET$DAY)

desc_df(cleandataset)
n mean sd median trimmed min max range skew kurtosis se %NA Q.1% Q.25% Q.75% Q.99%
T_MAX 16199 27.5756 4.8469 28.49 27.9438 12.590 40.30 27.710 -0.6797 0.1913 0.0381 0 14.3494 24.83 30.950 36.6404
T_MIN 16199 17.5620 6.1932 18.88 18.2900 -4.080 27.31 31.390 -0.8820 0.0307 0.0487 0 0.7700 13.85 22.870 25.3602
T_AVG 16199 22.5688 5.3084 23.90 23.1589 4.668 33.17 28.502 -0.8842 0.0769 0.0417 0 8.2897 19.50 26.745 30.2354
Show code 
cleandataset %>% 
  ggplot(aes(x=index(.)))+
  geom_line(aes(y=T_MAX, color = "T_MAX"))+
  geom_line(aes(y=T_MIN, color = "T_MIN"))+
  geom_line(aes(y=T_AVG, color = "T_AVG"))+
  labs(title = "Last 43 years of recorded data", y="Temperature", x=NULL)+
  scale_color_manual(name = "Temps", 
                     values = c(T_MAX = "indianred3",T_MIN = "lightblue",T_AVG = "lightgreen"))

However this is not that useful so let me zoom in on the last third of the database (approximately 2014 onward)

Show code 
NDAYS <- nrow(cleandataset)
lookback <- 365*10

tail(cleandataset, lookback) %>% 
  as.data.frame() %>% 
  mutate(year = index(tail(cleandataset, lookback))) %>% 
  ggplot(aes(x = year))+
  geom_line(aes(y=T_MAX, color = "T_MAX"))+
  geom_line(aes(y=T_MIN, color = "T_MIN"))+
  geom_line(aes(y=T_AVG, color = "T_AVG"))+
  labs(title = "Last 10 years of recorded data", y="Temperature", x=NULL)+
  scale_color_manual(name = "Temps", 
                    values = c(T_MAX = "indianred3",T_MIN = "lightblue",T_AVG = "lightgreen"))

Show code 
# library(gganimate)
# animplot1 <- plot1 +
#   labs(subtitle = "year: {frame_time}") +
#   transition_reveal(year) + 
#   view_follow(fixed_y = TRUE)
# 
# animate(animplot1, nframes = 20, renderer = gifski_renderer(), fps = 30, duration = 10, end_pause = 60, res = 100)

2.2 Seasonal analysis

Now i can look at the distribution of my database, first i need to distinguish the data between summer periods and winter periods by taking the day of the year and splitting it so that

Winter ~ October-March

Summer ~ April-September

Show code 
 DATASET_seas <- DATASET %>%
  group_by(Month < 4|Month>9) %>%
  rename("Season"="Month < 4 | Month > 9") %>%
  mutate(Season = ifelse(Season, "Winter", "Summer"))
  
DATASET_seas[-1] %>% 
  xts(order.by = DATASET$DAY)%>% 
  show_df() %>% 
  gt() %>%
  tab_header(title = "Temperature Overview") %>%
  opt_stylize(style = 5, add_row_striping = TRUE) %>%
  cols_align(align = "center") %>%
  sub_missing(columns = everything(), missing_text = "⋮")
Temperature Overview
DATE YEAR Month DOY T_MAX T_MIN T_AVG Season
1981-01-01 1981 1 1 18.96000 3.7600000 11.360000 Winter
1981-01-02 1981 1 2 14.78000 4.7100000 9.745000 Winter
1981-01-03 1981 1 3 17.66000 1.0000000 9.330000 Winter
1981-01-04 1981 1 4 19.79000 2.9600000 11.375000 Winter
1981-01-05 1981 1 5 14.88000 4.4500000 9.665000 Winter
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
2025-05-04 2025 5 124 31.01000 21.9100000 26.460000 Summer
2025-05-05 2025 5 125 32.31000 20.0000000 26.155000 Summer
2025-05-06 2025 5 126 34.18000 22.2700000 28.225000 Summer
2025-05-07 2025 5 127 34.48000 23.2600000 28.870000 Summer
2025-05-08 2025 5 128 33.77000 22.3900000 28.080000 Summer
Show code 
winter_dataset <- DATASET_seas[DATASET_seas$Season=="Winter",]
summer_dataset <- DATASET_seas[DATASET_seas$Season=="Summer",]

cat(paste0("The difference in number of rows is approximately: ", round(nrow(summer_dataset) / nrow(winter_dataset) - 1, 4)*100,"%", ", or ", round((nrow(summer_dataset) / nrow(winter_dataset) - 1)*NDAYS, 1), " days"))
The difference in number of rows is approximately: -0.23%, or -38 days
Show code 
grid.arrange(ncol=2,
ggplot(winter_dataset)+
  geom_histogram(aes(x = T_MAX, fill = "T_MAX"), alpha=0.8, bins = 80)+
  geom_histogram(aes(x = T_MIN, fill = "T_MIN"), alpha=0.8, bins = 80)+
  geom_histogram(aes(x = T_AVG, fill = "T_AVG"), alpha=0.8, bins = 80)+
  labs(title = "Distribution charts of winter",x=NULL)+
  scale_fill_manual(name = NULL, 
                     values = c(T_MAX = "indianred3",T_MIN = "lightblue",T_AVG = "lightgreen"))+
  theme(legend.position = "bottom")

,

ggplot(summer_dataset)+
  geom_histogram(aes(x = T_MAX, fill = "T_MAX"), alpha=0.8, bins = 80)+
  geom_histogram(aes(x = T_MIN, fill = "T_MIN"), alpha=0.8, bins = 80)+
  geom_histogram(aes(x = T_AVG, fill = "T_AVG"), alpha=0.8, bins = 80)+
  labs(title = "Distribution charts of summer",x=NULL)+
  scale_fill_manual(name = NULL, 
                    values = c(T_MAX = "indianred3",T_MIN = "lightblue",T_AVG = "lightgreen"))+
  theme(legend.position = "bottom")

)

Its with this choice of season im still able to keep the data very balanced. clear to see that there

Lastly I’m able to plot just the two average temperatures of winter and of summer for the average temperatures

Show code 
ggplot()+
  geom_histogram(data = winter_dataset, aes(x = T_AVG, fill = "Winter"), alpha=0.8, bins = 80)+
  geom_histogram(data = summer_dataset, aes(x = T_AVG, fill = "Summer"), alpha=0.8, bins = 80)+
  labs(title = "Distribution charts of the 2 averages",x=NULL)+
    scale_fill_manual(name = NULL, 
                    values = c(Winter="steelblue", Summer="orange"))+
  theme(legend.position = "bottom")

Just to understand the records of my dataset i wanted to see what months on record had the hottest and coldest days across all dataset

Show code 
DATASET_seas %>% 
  group_by(Month) %>% 
  summarise(T_min_min = min(T_MIN),
            T_min_max = max(T_MIN),
            T_max_min = min(T_MAX),
            T_max_max = max(T_MAX)) %>% 
  round(3) %>%
  mutate(Month = month.abb) %>% 
  gt() %>% 
  opt_stylize(5) %>% 
  tab_header("Data exploration", "what is the highest and lowest in my database") %>% 
  cols_label(T_min_min = "Min",
             T_min_max = "Max",
             T_max_min = "Min",
             T_max_max = "Max") %>% 
  tab_spanner(label = "T_MIN", columns = 2:3) %>%  
  tab_spanner(label = "T_MAX", columns = 4:5)
Data exploration
what is the highest and lowest in my database
Month
T_MIN
T_MAX
Min Max Min Max
Jan -4.08 20.90 12.59 29.78
Feb -3.58 20.24 12.62 32.49
Mar -1.26 22.43 13.21 35.29
Apr 1.63 23.82 14.38 36.70
May 9.48 25.50 21.34 39.12
Jun 13.06 26.49 24.94 40.30
Jul 19.55 27.13 26.31 38.58
Aug 19.61 27.31 25.72 37.71
Sep 13.40 26.03 23.38 35.13
Oct 2.68 25.56 15.78 34.12
Nov 1.66 24.50 12.88 32.13
Dec -3.90 22.12 12.61 30.43

3 Seasonal decomposition

The first step in pricing temperature options is to decompose the time series into distinct components that each represent different underlying patterns. This process is essential for understanding the structure of the data and for developing predictive models. In basic temrs it can be expressed as the sum of three main components:\(y_t=T_t+S_t+e_t\), Here, \(T_t\) is the trend component that captures the long-term movement, \(S_t\) represents the seasonal variations (periodic patterns), and \(e_t\) refers to the residuals or noise in the data.

which in our case becomes \[\overline{T_t} = T_{trend} + T_{seasonal}+ Residuals\] The approach was to develop each component indipendently

[T_{trend} = a + bt]

:

The trend component could be as simple as a straight line or a moving average

This form captures the cyclical nature of temperature changes throughout the year, like daily or annual temperature fluctuations.

[ T_{seasonal} = (t + ) ]

\[ T_{seasonal} = \alpha \cdot \sin(\omega \cdot t + \theta) \]

  • \(\alpha\) represents the amplitude

  • \(\omega\) is the angular frequency (related to the period of the cycle)

  • \(\theta\) is the phase shift

And lastly the residuals are just: \[Residuals = T_t - T_{trend} + T_{seasonal}\]:

Recent research has suggested that partial Fourier decomposition can be effective in modeling temperature data, particularly by omitting the cosine component. This approach simplifies the seasonal model while still accurately representing the cyclical nature of temperature changes. The sinusoidal term is often sufficient to describe seasonal temperature variations, making the model both interpretable and computationally efficient.

Before this I used a denoising procedure using Gaussian Convolution Filter as it also enhances the effectiveness of the Fourier transformation by reducing noise, allowing you to identify the dominant frequencies or cycles in your temperature data more accurately.

  • Gaussian Convolution Filter:

    • A convolution is a mathematical operation used to blend or smooth data. It involves applying a kernel (a small array of weights) across the data to compute a weighted average. The Gaussian kernel is a specific type of convolution filter that assigns weights based on the Gaussian (normal) distribution, which is a bell-shaped curve.

    • The Gaussian kernel is widely used for smoothing because it gives more weight to the central points (closer to the middle) while gradually reducing the weights for points further away. This is mathematically expressed as:

\[ G(x)=\frac{1}{\sqrt{2π\sigma^2}}\cdot e^{-\frac{x^2}{2\sigma^2}} \]

  • Moving Window: The Gaussian kernel slides across each point in the time series. For each position, it computes a weighted sum where the weights are defined by the Gaussian kernel.

  • Local Averaging: The computed value represents a smoothed average of the points within the window’s range. Points closer to the center of the window contribute more heavily to the average than those farther away, as per the Gaussian weights.

  • Symmetric Filtering: using 2 sides makes this operation symmetric, applying the kernel both forward and backward across the data, ensuring the filtering effect is centered.

  • Conclusion and data inspection: in visual terms the denoised data should look smoother with slightly lower peaks

Show code 
temps <- DATASET
apply_convolution <- function(x, kernel) {
  # Use filter() from stats package to apply convolution
  filtered <- stats::filter(x, kernel, sides = 2)  # Use sides = 2 for symmetric filter
  return(filtered)
}

# DENOISED - convolution with a window of 90 days
kernel <- dnorm(-3:3)
data.frame("Gaussian_Kernel" = round(kernel, 10))
Gaussian_Kernel
0.0044318
0.0539910
0.2419707
0.3989423
0.2419707
0.0539910
0.0044318
Show code 
temps$Denoised <- apply_convolution(temps$T_AVG, kernel)
temps$Denoised <- na.fill(temps$Denoised, mean(temps$Denoised, na.rm = TRUE))

temps$Trend <- SMA(temps$Denoised, n = lookback)

library(nlme, quietly = T, warn.conflicts = F)

# Define the model
sin_component <- function(t, a, b, alpha, theta) {
  omega <- 2 * pi / 365.25
  a + b * t + alpha * sin(omega * t + theta)
}
omega <- 2 * pi / 365.25

# Fit model using non linear squares
temps$NUM_DAY <- 1:nrow(temps)

fit <- nls(Denoised ~ sin_component(NUM_DAY, a, b, alpha, theta),
           data = temps,
           start = list(a = 1, b = 0, alpha = 1, theta = 0))


# Get coefficients and confidence intervals for the model
params <- coef(fit)
confint_fit <- confint(fit)
Waiting for profiling to be done...
Show code 
temps$SEAS <- params["alpha"] * sin(omega * temps$NUM_DAY + params["theta"])
temps$TREND <- params["a"] + params["b"] * temps$NUM_DAY
temps$BAR <- temps$TREND + temps$SEAS
temps$RESID <-  temps$T_AVG - temps$TREND - temps$SEAS

just to be certain i looked at if the residuals and the fitted values matched the \(\overline T\) and \(Residuals\) by checking percentage of same values as the rounding decimals increase The logic is that if the data sets are anything alike (like 2 different models results for the same dataset) you should see how far in the rounding are the data sets similar

Show code 
cat("T_BAR VS fitted from the non linear squares \n")
T_BAR VS fitted from the non linear squares
Show code 
check_acc(temps$BAR, fitted(fit),15)
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

Show code 
cat("T_BAR VS fitted from the non linear squares \n")
T_BAR VS fitted from the non linear squares
Show code 
check_acc(temps$RESID, residuals(fit),15)
Scale for y is already present.
Adding another scale for y, which will replace the existing scale.

3.1 Model performance

for assessing my model performance I’m looking at the values and confidence intervals and looking at the Residual sum of squares and the mean absolute error

  • The RSS function calculates the Residual Sum of Squares, which is a measure of the discrepancy between the observed data and the model’s predictions. A lower RSS indicates a better fit of the model to the data, meaning that the model’s predictions are closer to the actual observed values.

  • The MAE function is the average of the absolute differences between the observed and predicted values. It gives an idea of how far, on average, the predictions are from the actual values, without considering the direction of the error. A lower MAE indicates that the model’s predictions are generally close to the observed data.

a :  21.801  CI ~normally [ 21.718 , 21.883 ]
b :  0  CI ~normally [ 0 , 0 ]
alpha :  -6.034  CI ~normally [ -6.092 , -5.975 ]
theta :  -5.008  CI ~normally [ -5.018 , -4.998 ]
  RSS model sine curve: 157805.5 
  MAE model fit: 2.35

3.2 Visualization of results

Show code 
# plot denoised 
ggplot(tail(temps, lookback), aes(x=DAY))+
  geom_point(aes(y = T_AVG, color = "Average"), size = 1)+
  geom_point(aes(y = Denoised, color = "Denoised"), size = 1)+
  scale_color_manual(name=NULL, values = c(Average = "royalblue", Denoised = "#ffcc00"))+
  labs(title = "Average temperature", x = "Date", y = "Temperature", 
      subtitle = "Before and after the gaussian convolution filter")

Show code 
temps_xts <- temps %>%
  select(T_AVG, Denoised, TREND, SEAS, RESID) %>%
  xts(order.by = temps$DAY) %>%
  tail(lookback)

# Plot seasonal decomposition from Avg to residuals
grid.arrange(nrow=5, top = paste0("Classical decomposition - last ",  lookback/365, " years"), 
            
            temps_xts$T_AVG %>% 
            quickplot(subtitle = "Average Temperature", show_legend = F, xlab = NULL, ylab = "Temps"),

            temps_xts$Denoised %>% 
            quickplot(subtitle = "Denoised", show_legend = F, xlab = NULL, ylab = "Temps"),

            temps_xts$TREND %>% 
            quickplot(subtitle = "Trend", show_legend = F, xlab = NULL, ylab = "Temps"),

            temps_xts$SEAS %>% 
            quickplot(subtitle = "Seasonal", show_legend = F, xlab = NULL, ylab = "Temps"), 
            
            temps_xts$RESID %>% 
            quickplot(subtitle = "Residuals", show_legend = F, xlab = NULL, ylab = "Temps"))

Show code 
# Plot original vs. fitted data
ggplot(temps, aes(x = DAY)) +
  geom_point(aes(y = T_AVG), color = 'royalblue', size = 0.5) +
  geom_line(aes(y = BAR), color = 'orange', linewidth=2) +
  labs(title = "Temperature Model Fit (all Observations)", y = "Temperature (deg C)")

3.2.1 Check for possible model degradation

Checking for model degradation across time by looking at the first and last 10 years to see if there is a meaningful shift in where the sine curve and the average temperature

Show code 
grid.arrange(nrow = 2, ncol = 2,
ggplot(temps %>% head(lookback), aes(x = DAY)) +
  geom_point(aes(y = T_AVG), color = 'royalblue', size = 0.5) +
  geom_line(aes(y = BAR), color = 'orange', linewidth=2) +
  labs(title = paste0("Temperature Model Fit (First ", lookback/365, " years)"),x=NULL, y = NULL),

ggplot(temps %>% head(lookback), aes(x = DAY)) +
  geom_line(aes(y = RESID), color = 'black', linewidth=0.5) +
  labs(title = paste0("Residuals (First ", lookback/365, " years)"),x=NULL, y = NULL),

ggplot(temps %>% tail(lookback), aes(x = DAY)) +
  geom_point(aes(y = T_AVG), color = 'royalblue', size = 0.5) +
  geom_line(aes(y = BAR), color = 'orange', linewidth=2) +
  labs(title = paste0("Temperature Model Fit (Last ", lookback/365, " years)"), x=NULL, y = NULL),

ggplot(temps %>% tail(lookback), aes(x = DAY)) +
  geom_line(aes(y = RESID), color = 'black', linewidth=0.5) +
  labs(title = paste0("Residuals (Last ", lookback/365, " years)"),x=NULL, y = NULL)
)

3.3 Residuals analysis and diagnostics

  1. Autocorrelation Function (ACF) Plot of Residuals:

    The ACF plot checks for any correlation between residuals at different lags. If residuals are uncorrelated (i.e., resemble white noise), all values should fall within the confidence bands, indicating no significant autocorrelation.

  2. Partial Autocorrelation Function (PACF) Plot of Residuals:

    The PACF plot displays the partial correlation of residuals with their own lagged values, considering the effect of any intermediate lags. Like the ACF plot, if the model fits well, the PACF values should also fall within the confidence bands, showing no significant partial autocorrelations.

  3. Normality Check using QQ Plot:

    The QQ plot assesses whether the residuals are normally distributed. Points should lie along the reference line; significant deviations from this line suggest that the residuals do not follow a normal distribution, which could imply model misspecification or the presence of outliers.

  4. Histogram of Residuals with Normal Curve:

    The histogram visualizes the distribution of residuals, and the overlaid normal curve helps assess normality. If the histogram closely follows the bell-shaped curve, it suggests normal residuals. The vertical line representing kurtosis indicates whether the residuals are more or less peaked than a normal distribution.

  5. Residuals vs. Fitted Values Plot:

    The plot checks for patterns in the residuals against fitted values. Ideally, the residuals should be randomly scattered around zero, indicating homoscedasticity (constant variance). Any visible pattern or trend suggests issues like non-linearity, heteroscedasticity, or omitted variables.

Show code 
grid.arrange(nrow = 2, 
# ACF 
ggAcf(temps$RESID, lag.max = 100)+
  labs(title = "ACF of Residuals", x = NULL, y = NULL),

# PACF
ggPacf(temps$RESID, lag.max = 100)+
  labs(title = "PACF of Residuals", x = NULL, y = NULL),

## Check normality of residuals using QQ plot
ggplot(temps, aes(sample = RESID)) +
  stat_qq(color="royalblue")+
  stat_qq_line(color = "black", linewidth = 0.4)+
  labs(title = "QQ plot", x="Theoretical Quantiles", y= "Observed Quantiles"),

## Check for heteroskedasticity or any pattern in residuals
ggplot(temps %>% tail(lookback), aes(x=BAR, y = RESID))+
  geom_point(size = 0.4)+
  geom_smooth(method = "lm")+
  labs(title = paste0("Residuals vs Fitted - last ",  lookback/365, " years"), x= "Fitted Values", y = "Residuals"))
`geom_smooth()` using formula = 'y ~ x'

Show code 
# Histogram with bell curve and kurtosis
ggplot(temps, aes(x = RESID)) +
  geom_histogram(aes(y = after_stat(density)), fill = "lightblue", bins = 30) +
  stat_function(fun = dnorm, args = list(mean = mean(temps$RESID), sd = sd(temps$RESID)), 
                color = "red", linewidth = 1.2) +
  # geom_vline(xintercept = skewness(temps$RESID)[1], linetype = "dashed", linewidth=1, color = "darkred")+
  labs(title = "Histogram of Residuals with Normal Curve", x = "Residuals", y = "Density", 
        # subtitle = "Vertical line is kurtosis"
        )

4 Ornstein-Uhlenbeck (OU) process

Temperature dynamics using a mean-reverting stochastic process like the Ornstein-Uhlenbeck (OU) process, incorporate the seasonal adjustment into the drift rate to ensure that the expected value of the temperature follows the seasonal mean temperature.

The Ornstein-Uhlenbeck (OU) process is a type of stochastic (random) process that is often used in physics, finance, and other fields to model systems that exhibit some form of “mean-reverting” behavior. Let’s start from the basics to understand what this means.

A stochastic process is a collection of random variables representing the evolution of a system over time. In simpler terms, it describes how something changes when randomness is involved. For example, the price of a stock or the position of a particle under the influence of random forces can be modeled as a stochastic process.

\[ dX_t = \kappa (\mu−X_t) \ d_t + \sigma \cdot dW_t \]

  • \(X_t\) is the value of the process at time t.
  • \(\mu\) is the long-term mean or equilibrium level toward which the process reverts.
  • \(\theta>0\) is the rate of mean reversion, determining how fast the process reverts to the mean P.
  • \(\sigma >0\) is the volatility or the intensity of the random fluctuations.
  • \(dW_t\) is an infinitesimal increment of a Wiener process (Brownian motion).

Mean Reversion Term: \(\kappa (\mu−X_t) \ d_t\) : Representing the tendency of the process to revert to the mean \(\mu\). If \(X_t\)is above the mean \(\mu\), this term will be negative (pulling it down toward the mean). If \(X_t\) is below \(\mu\), the term will be positive (pushing it up toward the mean).

The speed of this pull or push is determined by \(\kappa\). A higher \(\kappa\) means the process will revert to the mean faster.

Random Fluctuation Term \(\sigma \cdot dW_t\) : represents the random noise or shock. This is a source of randomness, and it causes the process to deviate randomly over time.

The size of the noise is controlled by \(\sigma\), which is the volatility parameter.

Show code 
temps_OU <- temps
# Define parameters for the OU process
kappa <- 1-arima(temps_OU$RESID, order = c(1,0,0))$coef[1]  # Mean-reversion rate
sigma <- 0.1                                                # Volatility of the process
dt <- 1                                                     # Time step (daily data)

cat("Kappa is estimated as:", round(kappa,4))
Kappa is estimated as: 0.2431
Show code 
# Initialize variables for simulation
n <- nrow(temps_OU)                      # Number of time points
T_simulated <- numeric(n)                # Simulated temperature vector
T_simulated[1] <- temps_OU$Denoised[1]   # Set initial temperature to the first observed value

# Simulate the seasonal mean as a time-varying mean (trend + seasonal component)
T_bar <- temps_OU$BAR

# Simulate the modified OU process
for (i in 2:n) {
  # Rate of change of the seasonal mean
  dT_bar_dt <- (T_bar[i] - T_bar[i - 1]) / dt
  
  # Brownian motion increment
  dWt <- rnorm(1, mean = 0, sd = sqrt(dt))
  
  T_simulated[i] <- T_simulated[i - 1] + 
                    (dT_bar_dt + kappa * (T_bar[i] - T_simulated[i - 1])) * dt + 
                    sigma * dWt
}

temps_OU$OU <- T_simulated

DATE <- tail(temps$DAY, lookback)

grid.arrange(layout_matrix = matrix(data = c(1,1,1,2), ncol = 1), left = "Temperature",
temps_OU %>%
  select(Denoised, OU, BAR) %>%
  tail(lookback) %>%
  ggplot(aes(x = DATE)) +
  geom_point(aes(y = Denoised, color = "Denoised"), size = 0.5) +
  geom_line(aes(y = OU, color = "Simulated"), linewidth = 0.8) +
  scale_color_manual(name = NULL, values = c(Denoised = "#ffcc00", Simulated = "darkgreen")) +
  labs(title = "Simulated Ornstein-Uhlenbeck Process for Temperature - Last 10 years", x = NULL, y = NULL)+
  theme(axis)+ 
  theme(axis.title.x = element_blank(),
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank())
,
temps_OU %>%
  select(Denoised, OU, BAR) %>%
  tail(lookback) %>%
  ggplot(aes(x = DATE)) +
  geom_line(aes(y = Denoised-OU, color = "Difference"), linewidth = 0.5) +
  scale_color_manual(name = NULL, values = c(Difference = "mediumspringgreen")) +
  labs(x = "Day", y = NULL)
)

5 Modeling volatility

5.1 Polynomial Regression:

Fit polynomial curves to model the relationship between day-of-year and volatility.

Show code 
# Create a dataframe with temperature and date components
temp_t <- data.frame(
  Date = temps$DAY,  # Assuming the index is a proper Date object
  T = temps$Denoised,
  day = yday(temps$DAY),
  month = month(temps$DAY)
)

# 2. Calculate monthly volatility statistics
vol1 <- temps %>%
  group_by(YEAR, Month) %>%
  summarise(
    mean_temp = mean(Denoised, na.rm = TRUE),
    std_temp = sd(Denoised, na.rm = TRUE)
  ) %>%
  ungroup()
`summarise()` has grouped output by 'YEAR'. You can override using the
`.groups` argument.
Show code 
# 5. Volatility Analysis
    vol <- temps %>%
      select(DAY, T_AVG) %>%
      mutate(day = yday(DAY),
            month = month(DAY)) %>%
      group_by(day) %>%
      summarise(mean = mean(T_AVG, na.rm = TRUE),
                std = sd(T_AVG, na.rm = TRUE))

5.2 B-splines

Flexible non-parametric method that fits piecewise ials (splines) at different intervals (knots).

Avoids overfitting by controlling the number of knots.

AIC (Akaike Information Criterion): Helps select the best model by balancing goodness-of-fit and complexity (lower AIC preferred).

The residual sum of squares (RSS) measures the level of variance in the error term, or residuals, of a regression model. The smaller the residual sum of squares, the better your model fits your data; the greater the residual sum of squares, the poorer your model fits your data.

Show code 
library(splines)
x <- 1:366
y <- vol$std

# Define the number of knots
knots <- c(1, 3, 5, 10, 15, 20, 30, 50, 80)
x <- 1:366
# Function to create a spline model and plot
create_spline_plot <- function(knots, x, y) {
  # Fit the spline model
  spline_model <- lm(y ~ bs(x, knots = knots))
  yfit <- predict(spline_model, data.frame(x = x))
  
  # Calculate Residual Sum of Squares (RSS)
  rss <- sum((y - yfit)^2)
  aic <- length(x) * log(rss / length(x)) + 2 * knots

  # Create the plot
  plots <- ggplot() +
    geom_point(aes(x, y), color = 'cornflowerblue', size = 1.5) +
    geom_line(aes(x, yfit), color = 'black', linewidth = 1) +
    ggtitle(paste("Knots #", knots, "\nRSS:", round(rss, 2), "AIC:", round(aic,2))) +
    labs(x = NULL, y = NULL)
    theme_classic()+
    theme(plot.title = element_text(size = 10, face = "bold"))
  
  return(plots)
}

# Generate all the plots
plots <- lapply(knots, create_spline_plot, x = x, y = y)

# Arrange and display the plots in a 2x3 grid
grid.arrange(grobs = plots, nrow = 3, ncol = 3, left = "Standard deviation of temperatures", bottom = "Day of the year", top = "Different B-splines models")

5.3 Polynomial models to fit volatility

Show code 
# Function to create polynomial model and plot
x <- 1:366
create_poly_plot <- function(degree, x, y) {
  # Fit polynomial model
  model <- lm(y ~ poly(x, degree, raw = TRUE))
  yfit <- predict(model, data.frame(x = x))
  
  # Calculate metrics
  rss <- sum(resid(model)^2)
  aic <- AIC(model)
  
  # Create plot
  ggplot() +
    geom_point(aes(x, y), color = '#00962B', size = 1) +
    geom_line(aes(x, yfit), color = 'black', linewidth = 1) +
    ggtitle(paste("Degree:", degree, "\nRSS:", round(rss, 2), "AIC:", round(aic, 2))) +
    labs(y = NULL, x = NULL)+
    theme_classic()+
    theme(plot.title = element_text(size = 10, face = "bold"))
  }

# Generate plots for degrees 1 through 9
degrees <- 1:9
plots <- lapply(degrees, create_poly_plot, x = x, y = y)

# Arrange in 3x3 grid
grid.arrange(grobs = plots, nrow = 3, ncol = 3, left = "Standard deviation of temperatures", bottom = "Day of the year", top = "Different polynomial models")

5.4 Fourier transforms

Show code 
fourier_series <- function(x, n_terms, period = 365.25) {
  omega <- 2 * pi / period
  terms <- list(a0 = 1)

  for (i in 1:n_terms) {
    terms[[paste0("a", i)]] <- cos(i * omega * x)
    terms[[paste0("b", i)]] <- sin(i * omega * x)
  }
  return(terms)
}

fourier_series(1:366, 3) %>%
  data.frame("actual" = normalize(y, 1)) %>%
  quickplot(
    title = "Fourier transformation different components with actual volatility",
    subtitle = "Order 3 so 7 components",
    xlab = "Day of the year"
  )

Show code 
create_fourier_plot <- function(n_terms, x, y) {
  # Fit Fourier series
  fourier_terms <- fourier_series(1:366, n_terms = n_terms)
  model_data <- data.frame(y = y, fourier_terms)

  fourier_fit <- lm(y ~ ., data = model_data)
  fourier <- predict(fourier_fit, newdata = model_data)

  # Calculate metrics
  rss <- sum(resid(fourier_fit)^2)
  aic <- AIC(fourier_fit)

  # Create plot
  ggplot(data = data.frame(), aes(x = x)) +
    geom_point(aes(y = y), color = 'orangered3', size = 1) +
    geom_line(aes(y = fourier), color = 'black', linewidth = 1) +
    ggtitle(paste("Order:", n_terms, "\nRSS:", round(rss, 2), "AIC:", round(aic, 2))) +
    labs(y = NULL, x = NULL) +
    theme_classic() +
    theme(plot.title = element_text(size = 10, face = "bold"))
}

# Generate plots for degrees 1 through 9
order <- c(0:8)
plots <- lapply(order, create_fourier_plot, x = x, y = y)

# Arrange in 3x3 grid
grid.arrange(grobs = plots, nrow = 3, ncol = 3,
  left = "Standard deviation of temperatures",
  bottom = "Day of the year",
  top = "Different Fourier Series models")

5.5 Regime switch

Show code 
library(changepoint)
Successfully loaded changepoint package version 2.3
 WARNING: From v.2.3 the default method in cpt.* functions has changed from AMOC to PELT.
 See NEWS for details of all changes.

Attaching package: 'changepoint'
The following object is masked from 'package:rugarch':

    likelihood
Show code 
create_changepoint_plot <- function(n_breaks, x, y) {
  # Detect changepoints in variance with fixed number of breaks
  cp <- cpt.var(y, method = "PELT", Q = n_breaks)

  # Get regime-wise standard deviation
  regime_sd <- rep(NA, length(y))
  segs <- c(1, cpts(cp), length(y))
  for (i in 1:(length(segs) - 1)) {
    idx <- segs[i]:(segs[i + 1] - 1)
    regime_sd[idx] <- sd(y[idx])
  }

  # RSS (within segment variance sum)
  rss <- sum((y - na.omit(regime_sd))^2)

  # Build plot
  data_plot <- data.frame(x = x, y = y, sd_fit = regime_sd)
  ggplot(data_plot, aes(x = x)) +
    geom_point(aes(y = y), color = 'purple4', size = 1) +
    geom_line(aes(y = sd_fit), color = 'black', linewidth = 1) +
    ggtitle(paste("Breaks:", n_breaks, "\nRSS:", round(rss, 2))) +
    theme_classic() +
    theme(plot.title = element_text(size = 10, face = "bold"))
}

# Create 9 plots for rhe breaks
breaks <- c(1, 2, 4, 8, 10, 12, 20, 30, 40)
plots <- lapply(breaks, create_changepoint_plot, x = x, y = y)
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Warning in y - na.omit(regime_sd): longer object length is not a multiple of
shorter object length
Show code 
grid.arrange(grobs = plots, nrow = 3, ncol = 3,
  left = "Standard deviation of temperatures",
  bottom = "Day of the year",
  top = "Variance Regime Switching Models")
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).
Removed 1 row containing missing values or values outside the scale range
(`geom_line()`).

Show code 
# 3. Print long-term volatility metrics
cat("Trend or long term volatility is easy: ~", round(mean(vol1$std_temp, na.rm = TRUE), 3), "\n")
Trend or long term volatility is easy: ~ 2.024
Show code 
cat("Gamma is:", round(sd(vol1$std_temp, na.rm = TRUE), 3), "\n")
Gamma is: 1.125
Show code 
# 4. Fit AR(1) model for mean reversion rate
ar_model <- arima(vol1$std_temp, order = c(1, 0, 0), include.mean = FALSE)
coef <- ar_model$coef
residuals <- ar_model$residuals

cat("Rate of mean reversion of volatility process is:", round(coef["ar1"], 3), "\n")
Rate of mean reversion of volatility process is: 0.917
Show code 
print(summary(ar_model))

Call:
arima(x = vol1$std_temp, order = c(1, 0, 0), include.mean = FALSE)

Coefficients:
         ar1
      0.9173
s.e.  0.0172

sigma^2 estimated as 0.8632:  log likelihood = -718,  aic = 1440

Training set error measures:
                    ME      RMSE       MAE       MPE    MAPE      MASE
Training set 0.1660121 0.9290675 0.6942639 -4.485931 39.3609 0.9684978
                   ACF1
Training set -0.1142199

5.5.1 IDK

This code models temperature data using a linear trend plus seasonal component, then analyzes the residuals.

The model combines:

  • A linear trend (\(a+bx\))

  • A seasonal sine wave ( \(\alpha sin(\omega x + \theta)\)), where:

\(\alpha\) = amplitude (how strong the seasonality is)

\(\theta\) = phase shift (when peaks/troughs occur)

\(\omega\) = fixed frequency ($2π/365.25$ to match yearly cycles)

Model Fitting: Uses nlsLM() from the minpack.lm package for Levenberg-Marquardt fitting

  • Uses nonlinear least-squares (nlsLM) to fit:

    • \(a\) = intercept (baseline temperature)

    • \(b\) = slope (long-term warming/cooling trend)

    • \(a_1\), \(b_1\) = sine/cosine coefficients (combined into \(\alpha\) and \(\theta\))

Show code 
library(minpack.lm)  # For Levenberg-Marquardt nonlinear least squares

# Define the model functions
model <- function(x, params) {
  a <- params[1]
  b <- params[2]
  a1 <- params[3]
  b1 <- params[4]
  
  omega <- 2*pi/365.25
  theta <- atan2(a1, b1)
  alpha <- sqrt(a1^2 + b1^2)
  
  cat("Parameters:\n")
  cat(sprintf("     a %0.3f\n", a))
  cat(sprintf("     b %0.3f\n", b))
  cat(sprintf(" alpha %0.3f\n", alpha))
  cat(sprintf(" theta %0.3f\n", theta))
  
  y_pred <- a + b*x + alpha*sin(omega*x + theta)
  return(y_pred)
}

model_fit <- function(x, a, b, a1, b1) {
  omega <- 2*pi/365.25
  y_pred <- a + b*x + a1*cos(omega*x) + b1*sin(omega*x)
  return(y_pred)
}

# Prepare the data
first_ord <- as.numeric(temp_t$Date[1])
x_data <- as.numeric(temp_t$Date) - first_ord
y_data <- temps$T_AVG

# Fit the model using nonlinear least squares
fit <- nlsLM(
  y_data ~ model_fit(x_data, a, b, a1, b1),
  start = list(a = mean(y_data), b = 0, a1 = 0, b1 = 0),
  control = nls.lm.control(maxiter = 100)
)

# Extract parameters
params_all <- coef(fit)
temps$model <- model(x_data, params_all)
Parameters:
     a 21.796
     b 0.000
 alpha 6.041
 theta -1.849
Show code 
temp_t$res <- temps$T_AVG - temps$model

# Plot the last 2000 observations
temps %>%
  tail(2000) %>%
  ggplot(aes(x = DAY)) +
    geom_point(aes(y = T_AVG, color = "Observed"), size = 2) +
    geom_line(aes(y = model, color = "Model"), linewidth = 1) +
    scale_color_manual(values = c("Observed" = "blue", "Model" = "darkorange")) +
    labs(y = "Temperature", title = "Temperature Model Fit") +
    theme_minimal() +
    theme(legend.position = "bottom")

Show code 
# Model performance
cat("  RSS model sine curve:", round(RSS(temps$T_AVG, temps$model), 2), "\n")
  RSS model sine curve: 157804
Show code 
cat("  MAE model fit:", round(MAE(temps$model, temps$T_AVG), 2), "\n")
  MAE model fit: 2.35

This model:

  1. Detects long-term trends (climate change via \(b\))

  2. Quantifies seasonality (summer/winter differences via \(\alpha\))

  3. Identifies anomalies (via residuals res), useful for:

    • Weather derivative pricing

    • Volatility analysis (like your earlier code) # Montecarlo simulations

modified ou model includes the time drift component

\[ dT_t=\left(\frac{dT_{bar}(t)}{dt}+κ(T_{bar}(t)−T_t)\right)dt + σ(t)\ dW_t \]

Show code 
a <- params[1]
b <- params[2]
theta <- atan2(params[3], params[4])
alpha <- sqrt(params[3]^2 + params[4]^2)
kappa <- as.double(kappa)

data.frame(a = a, b = b, theta = theta, alpha = alpha, kappa = kappa)
a b theta alpha kappa
a 21.80081 9.75e-05 -2.263556 7.841268 0.2430516
Show code 
# 1. Temperature Model Functions
T_model <- function(x, a, b, alpha, theta) {
  omega <- 2 * pi / 365.25
  a + b * x + alpha * sin(omega * x + theta)
}
dT_model <- function(x, a, b, alpha, theta) {
  omega <- 2 * pi / 365.25
  b + alpha * omega * cos(omega * x + theta)
}

# 2. Prepare Data
if (inherits(temps$DAY, "Date")) {
  first_ord <- as.numeric(temps$DAY[1])
  temp_t$ordinal <- as.numeric(temps$DAY)
} else {
  temp_t$Date <- as.Date(temps$DAY)
  first_ord <- as.numeric(temps$DAY[1])
  temp_t$ordinal <- as.numeric(temps$DAY)
}

# 3. Apply Model with Given Parameters
Tbar_params <- list(a = a, b = b, alpha = alpha, theta = theta)

temps$model_fit <- T_model(
  temp_t$ordinal - first_ord,
  Tbar_params$a,
  Tbar_params$b,
  Tbar_params$alpha,
  Tbar_params$theta
)

grid.arrange(
  nrow = 2,
  ncol = 2,
  ggplot(temps %>% head(lookback), aes(x = DAY)) +
    geom_point(aes(y = T_AVG), color = 'royalblue', size = 0.5) +
    geom_line(aes(y = model_fit), color = 'orange', linewidth = 2) +
    labs(
      title = paste0(
        "Temperature Model Fit (First ",
        lookback / 365,
        " years)"
      ),
      x = NULL,
      y = NULL
    ),

  ggplot(temps %>% head(lookback), aes(x = DAY)) +
    geom_line(aes(y = T_AVG - model_fit), color = 'black', linewidth = 0.5) +
    labs(
      title = paste0("Residuals (First ", lookback / 365, " years)"),
      x = NULL,
      y = NULL
    ),

  ggplot(temps %>% tail(lookback), aes(x = DAY)) +
    geom_point(aes(y = T_AVG), color = 'royalblue', size = 0.5) +
    geom_line(aes(y = model_fit), color = 'orange', linewidth = 2) +
    labs(
      title = paste0("Temperature Model Fit (Last ", lookback / 365, " years)"),
      x = NULL,
      y = NULL
    ),

  ggplot(temps %>% tail(lookback), aes(x = DAY)) +
    geom_line(aes(y = T_AVG - model_fit), color = 'black', linewidth = 0.5) +
    labs(
      title = paste0("Residuals (Last ", lookback / 365, " years)"),
      x = NULL,
      y = NULL
    )
)

Show code 
# 6. Spline Fit for Volatility
spline_fit <- function(knots, x, y) {
  x_new <- seq(0, 1, length.out = knots + 2)[2:(knots + 1)]
  knots_pos <- quantile(x, probs = x_new)
  bspline <- lm(y ~ bs(x, knots = knots_pos, degree = 15))
  predict(bspline, newdata = data.frame(x = x))
}

volatility <- spline_fit(15, vol$day, vol$std)

# Plot Volatility
ggplot(vol, aes(x = day)) +
  geom_point(aes(y = std, color = "Observed Volatility"), shape = 8) +
  geom_line(aes(y = volatility, color = "Spline Fit"), linewidth = 1) +
  scale_color_manual(
    values = c("Observed Volatility" = "blue", "Spline Fit" = "black")
  ) +
  labs(
    title = "Temperature Volatility by Day of Year",
    y = "Std Dev (°C)",
    x = "Day of Year"
  ) +
  theme_minimal()

Show code 
# 7. Monte Carlo Simulation Functions
euler_step <- function(row, kappa, M) {
  T_i <- ifelse(is.na(row$Tbar_shift), row$Tbar, row$Tbar_shift)
  T_det <- T_i + row$dTbar
  T_mrev <- kappa * (row$Tbar - T_i)
  sigma <- row$vol * rnorm(M)
  T_det + T_mrev + sigma
}

monte_carlo_temp <- function(
  trading_dates,
  Tbar_params,
  vol_model,
  first_ord,
  M = 1,
  kappa = 0.2430516
) {
  # Convert dates to numeric if needed
  if (inherits(trading_dates, "Date")) {
    trading_numeric <- as.numeric(trading_dates)
  } else {
    trading_dates <- as.Date(trading_dates)
    trading_numeric <- as.numeric(trading_dates)
  }

  # Calculate Tbar and dTbar
  x_vals <- trading_numeric - first_ord
  Tbars <- T_model(
    x_vals,
    Tbar_params$a,
    Tbar_params$b,
    Tbar_params$alpha,
    Tbar_params$theta
  )
  dTbars <- dT_model(
    x_vals,
    Tbar_params$a,
    Tbar_params$b,
    Tbar_params$alpha,
    Tbar_params$theta
  )

  # Create simulation dataframe
  mc_temps <- data.frame(
    Date = trading_dates,
    Tbar = Tbars,
    dTbar = dTbars,
    day = yday(trading_dates),
    vol = vol_model[yday(trading_dates)] # Directly add volatility
  )

  # Add lagged Tbar
  mc_temps$Tbar_shift <- dplyr::lag(mc_temps$Tbar)

  # Run simulations - modified apply call
  simulations <- sapply(1:nrow(mc_temps), function(i) {
    row <- mc_temps[i, ]
    euler_step(row, kappa, M)
  })

  # Transpose and format results
  simulations <- t(simulations)
  colnames(simulations) <- paste0("Sim", 1:M)

  list(
    mc_temps = mc_temps,
    mc_sims = cbind(Date = trading_dates, as.data.frame(simulations))
  )
}

# 8. Run Simulation
trading_dates <- seq(as.Date("2022-09-01"), as.Date("2025-08-31"), by = "day")
sim_results <- monte_carlo_temp(
  trading_dates,
  Tbar_params,
  volatility,
  first_ord,
  M = 5
)

# 9. Plot Simulation Results
sim_results$mc_sims %>%
  pivot_longer(-Date, names_to = "Simulation", values_to = "Temperature") %>%
  ggplot(aes(x = Date, y = Temperature, color = Simulation)) +
  geom_point(alpha = 0.7) +
  labs(title = "Monte Carlo Temperature Simulations", y = "Temperature (°C)") +
  theme_minimal() +
  theme(legend.position = "bottom")

Show code 
# Set number of simulations
no_sims <- 100000

# Define winter and summer dates (Southern Hemisphere)
trading_dates_winter <- as.Date("2025-10-01")
trading_dates_summer <- as.Date("2025-04-01")

# Run simulations
sim_results_winter <- monte_carlo_temp(trading_dates_winter, Tbar_params, volatility, first_ord, M = no_sims)
sim_results_summer <- monte_carlo_temp(trading_dates_summer, Tbar_params, volatility, first_ord, M = no_sims)


# Extract results
mc_sims_winter <- sim_results_winter$mc_sims %>% select(-Date)
mc_sims_summer <- sim_results_summer$mc_sims %>% select(-Date)

Tbar_summer <- sim_results_summer$mc_temps$Tbar[1]
Tbar_winter <- sim_results_winter$mc_temps$Tbar[1]

# Create combined data frame for plotting
plot_data <- bind_rows(
  data.frame(Temperature = unlist(mc_sims_summer), Season = "Summer"),
  data.frame(Temperature = unlist(mc_sims_winter), Season = "Winter")
)

# Create the plot
ggplot(plot_data, aes(x = Temperature, fill = Season)) +
  geom_histogram(position = "identity", alpha = 0.8, bins = 80) +
  geom_vline(aes(xintercept = Tbar_winter), color = "darkblue",
              linewidth = 1.5, linetype = "solid") +
  geom_vline(aes(xintercept = Tbar_summer), color = "darkorange",
              linewidth = 1.5, linetype = "solid") +
  scale_fill_manual(values = c(Winter="steelblue", Summer="orange")) +
  labs(title = "Winter vs Summer Temperature MC Sims",
        x = "Temperature (°C)", 
        y = "Frequency") +
  theme_minimal() +
  theme(legend.position = "bottom")

6 Sidequests

6.1 Sidequest: overlap of the dataset

One of the points that i would need for later is the standard deviation across the days of year, a pivot table is used for this step as with this I’m able to line up all the dataset month to month, by doing this im also able to analize seasonal patterns and look at the seasonality at a glance

Show code 
pivot_df <- DATASET %>%
  select(DOY, YEAR, T_AVG) %>%
  pivot_wider(names_from = YEAR, values_from = T_AVG)


MAX_pivot_df <- DATASET %>%
  select(DOY, YEAR, T_MAX) %>%
  pivot_wider(names_from = YEAR, values_from = T_MAX)

MIN_pivot_df <- DATASET %>%
  select(DOY, YEAR, T_MIN) %>%
  pivot_wider(names_from = YEAR, values_from = T_MIN)


MEAN <- apply(pivot_df[-1], 1, mean, na.rm=TRUE)
MEDIAN <- apply(pivot_df[-1], 1, median, na.rm=TRUE)
IQR <- apply(pivot_df[-1], 1, IQR, na.rm=TRUE)
SD <- apply(pivot_df[-1], 1, sd, na.rm=TRUE)

data.frame(MEAN = MEAN,
          MEDIAN = MEDIAN) %>% 
  quickplot(title = "Mean and median across the year", xlab = "Day of the year", ylab = "Temperature")

Show code 
data.frame(IQR = SMA(IQR, n = 7),
          SD = SD) %>% 
  quickplot(title = "Standard deviation and Inter-quartile-range across the year",
            subtitle = "IQR MA(7)", xlab = "Day of the year", ylab = "Temperature")
Warning: Removed 6 rows containing missing values or values outside the scale range
(`geom_line()`).

Show code 
data.frame(MAX = apply(pivot_df[-1], 1, max, na.rm=TRUE), 
          MIN = apply(pivot_df[-1], 1, min, na.rm=TRUE)) %>%
  mutate(AVG = (MAX + MIN)/2) %>% 
  mutate(RANGE = MAX - MIN) %>% 
  quickplot(title = "Range across the year", show_legend = T, xlab = "Day of the year", ylab = "Temperature")

Show code 
melt(pivot_df[-1], id.vars = NULL) %>% 
  ggplot(aes(x = variable, y = value)) +
    geom_boxplot(fill = "gray") +
    theme(axis.text.x = element_text(angle = 45, hjust = 1)) +  
    labs(title = "Boxplot for all years", x = NULL, y = NULL)
Warning: Removed 271 rows containing non-finite outside the scale range
(`stat_boxplot()`).

Show code 
DATASET %>%
  select(Month, T_AVG) %>%
  group_by(Month) %>% 
  ggplot(aes(x = Month, y = T_AVG, group = Month)) +
    geom_boxplot(fill = "gray") +
    scale_x_continuous(breaks = seq(1, 12, by = 1), labels = month.abb) +
    labs(title = "Boxplot for all months across the years", x = NULL, y = NULL)

6.1.1 How is this year compared to the rest

this dataset is 43 years long and there have been some really hot points and some really cold points over the years, so i wanted to look at my dataset and create a ribbon of all the max and mins, to see if this year has been so hot when looking at the averages, while this is in no way conclusive evidence it at least gives you a glimpse at how the climate is changing especially when looking at the middle of the year.

Show code 
ggplot(DATASET, aes(x = DOY, y = T_AVG, group = YEAR, color = YEAR)) +
  geom_line() +
  labs(title = "All years overlapped", x = "Day of the year", y = "Index level of returns")

Show code 
THISYEAR <- data.frame(
  c(DATASET[DATASET$YEAR == 2024, ]["T_MAX"]),
  c(DATASET[DATASET$YEAR == 2024, ]["T_AVG"]),
  c(DATASET[DATASET$YEAR == 2024, ]["T_MIN"])
)

cbind(pivot_df,
      "MIN" = apply(pivot_df[-1], 1, min, na.rm = TRUE),
      "MAX" = apply(pivot_df[-1], 1, max, na.rm = TRUE),
      "MEAN" = apply(pivot_df[-1], 1, mean, na.rm = TRUE)
) %>%
  round(2) %>%
  ggplot(aes(x = DOY)) +
    geom_ribbon(aes(ymin = MIN, ymax = MAX), alpha = 0.2) +
    geom_line(aes(y = MEAN, color = "MEAN"), linewidth = 1) +
    geom_line(aes(y = pivot_df$`2025`, color = "AVG_Current"), linewidth = 1.3) +
    geom_line(aes(y = MAX_pivot_df$`2025`, color = "MAX_Current"), linewidth = 0.6, alpha = 1) +
    geom_line(aes(y = MIN_pivot_df$`2025`, color = "MIN_Current"), linewidth = 0.6, alpha = 1) +
  scale_color_manual(name = NULL, values = c(MEAN = "orange2", AVG_Current = "olivedrab", 
                                                MIN_Current = "lightblue", MAX_Current = "indianred")) +
  labs(title = "Is this year hotter on average?", y = NULL, x = NULL)+
  theme(legend.position = "bottom")
Warning: Removed 238 rows containing missing values or values outside the scale range
(`geom_line()`).
Removed 238 rows containing missing values or values outside the scale range
(`geom_line()`).
Removed 238 rows containing missing values or values outside the scale range
(`geom_line()`).

6.2 Sidequest: predicting temperatures in the future

Right now on 11-5-2025 i want to do an experiment and actually see if at least in some way i can predict the weather using this model that i’ve built right now i want to predict the temperature une month from now and

Show code 
model_formula <- function(t, a, b, a1, b1) {
  omega <- 2 * pi / 365.25
  a + b * t + a1 * cos(omega * t) + b1 * sin(omega * t)
}

forecast_N <- 30
new_t <- max(temps$NUM_DAY) + forecast_N  # forecast_N days after last observation
predicted_T <- model_formula(new_t, params["a"], params["b"], params["alpha"], params["theta"])
cat("predicted temperature for", as.character(max(temps$DAY)+forecast_N), "=", predicted_T)
predicted temperature for 2025-06-07 = 26.82483
Show code 
arima_model <- Arima(temps$RESID, order = c(2, 0, 0), include.mean = FALSE)
future_residuals <- forecast(arima_model, h = forecast_N)  # forecast_N steps ahead

first_date <- min(temps$DAY)
future_dates <- seq(max(temps$DAY), by = "day", length.out = forecast_N)
future_t <- as.numeric(difftime(future_dates, first_date, units = "days"))

# Deterministic part
deterministic_part <- model_formula(future_t, params["a"], params["b"], params["alpha"], params["theta"])

# Stochastic part (residuals)
stochastic_part <- future_residuals$mean

# Final prediction
future_T <- deterministic_part + stochastic_part

forecast_df <- data.frame(
  Date = future_dates,
  Temperature = future_T,
  Lower = future_T - 1.96 * future_residuals$mean,  # 95% CI
  Upper = future_T + 1.96 * future_residuals$mean
)

ggplot() +
  geom_line(data = tail(temps, 100), aes(x = DAY, y = T_AVG)) +
  geom_line(data = forecast_df, aes(x = Date, y = Temperature), color = "red") +
  geom_ribbon(data = forecast_df, aes(x = Date, ymin = Lower, ymax = Upper), alpha = 0.2) +
  labs(title = "Temperature Forecast with 95% CI", y = "Temperature (°C)", x = "Date")

The residuals represent short-term deviations from the deterministic trend/seasonality, becoming irrelevant after 15 days, which means that in the long run you will get a temperature entirely determined by the deterministic trend, over time, these deviations are expected to fade back to zero (the mean).

Forecast implications:

for forecasting purposes it means a couple of things: that in the short-term my Residuals adjust for recent “shocks” (e.g., a heatwave/cold snap), but in the long-term the forecast relies entirely on the deterministic component (trend + seasonality).

This model doesnt really account for Volatility clustering, like how temperature tends to have time-varying volatility (e.g., summer vs. winter variability). which is the part of the OU that we still need to model

Show code 
# Assuming trading_dates and sim_length already defined
x_vals <- as.numeric(trading_dates) - first_ord
Tbar_vals <- T_model(x_vals, a, b, alpha, theta)
dTbar_vals <- dT_model(x_vals, a, b, alpha, theta)
sigma_vals <- volatility[yday(trading_dates)]

data.frame(
  x_vals = x_vals,
  Tbar_vals = Tbar_vals,
  dTbar_vals = dTbar_vals,
  sigma_vals = sigma_vals
) %>% show_df()
DATE x_vals Tbar_vals dTbar_vals sigma_vals
1 15218 30.67273 -0.0450815 0.9961240
2 15219 30.62656 -0.0472611 0.9922418
3 15220 30.57822 -0.0494267 0.9902199
4 15221 30.52771 -0.0515776 0.9900876
5 15222 30.47507 -0.0537132 0.9918538
NA NA NA NA NA
1092 16309 30.96880 -0.0345569 1.0409072
1093 16310 30.93313 -0.0367942 1.0289168
1094 16311 30.89522 -0.0390206 1.0183175
1095 16312 30.85509 -0.0412354 1.0092453
1096 16313 30.81275 -0.0434380 1.0018160
Show code 
data.frame(
  x_vals = x_vals,
  Tbar_vals = Tbar_vals,
  dTbar_vals = dTbar_vals,
  sigma_vals = sigma_vals
) %>% apply(2, normalize) %>% quickplot()

Show code 
# library(sde)
# # Interpolate for arbitrary t
# drift_fun <- approxfun(seq_along(Tbar_vals), dTbar_vals + kappa * (Tbar_vals - Tbar_vals))  # Initially centered
# sigma_fun <- approxfun(seq_along(sigma_vals), sigma_vals)

# d <- expression(drift_fun(t))
# s <- expression(sigma_fun(t))

# X <- sde.sim(X0 = Tbar_vals[1], drift = d, sigma = s, N = length(trading_dates), T = 3)

# plot(X, type = "l", main = "SDE Simulated Seasonal Temperature")